Dimensional Optimal Control Problem Research Articles

Solving Two-Point Boundary Value Problems for a Wave Equation via the Principle of Stationary Action and Optimal Control

A new approach to solving two-point boundary value problems for a wave equation is developed. This new approach exploits the principle of stationary action to reformulate and solve such problems in the framework of optimal control. In particular, an infinite dimensional optimal control problem is posed so that the wave equation dynamics and temporal boundary data of interest are captured via the characteristics of the associated Hamiltonian and choice of terminal payoff, respectively. In order to solve this optimal control problem for any such terminal payoff, and hence solve any two-point boundary value problem corresponding to the boundary data encapsulated by that terminal payoff, a fundamental solution to the optimal control problem is constructed. Specifically, the optimal control problem corresponding to any given terminal payoff can be solved via a max-plus convolution of this fundamental solution with the specified terminal payoff. Crucially, the fundamental solution is shown to be a quadratic functional that is defined with respect to the unique solution of a set of operator differential equations, and computable using spectral methods. An example is presented in which this fundamental solution is computed and applied to solve a two-point boundary value problem for the wave equation of interest.

Finite dimensional approximations for a class of infinite dimensional time optimal control problems

ABSTRACTIn this work, we study the numerical approximation of the solutions of a class of abstract parabolic time optimal control problems with unbounded control operator. Our main results assert that, provided that the target is a closed ball centered at the origin and of positive radius, the optimal time and the optimal controls of the approximate time optimal problems converge (in appropriate norms) to the optimal time and to the optimal controls of the original problem. In order to prove our main theorem, we provide a nonsmooth data error estimate for abstract parabolic systems.

A leapfrog semi-smooth Newton-multigrid method for semilinear parabolic optimal control problems

A new semi-smooth Newton multigrid algorithm is proposed for solving the discretized first order necessary optimality systems that characterizing the optimal solutions of a class of two dimensional semi-linear parabolic PDE optimal control problems with control constraints. A new computational scheme (leapfrog scheme) in time associated with the standard five-point stencil in space is established to achieve the second-order finite difference discretization. The convergence (or unconditional stability) of the proposed scheme is proved when assuming time-periodic solutions. Moreover, the derived well-structured discretized Jacobian matrices greatly facilitate the development of effective smoother in our multigrid algorithm. Numerical simulations are provided to illustrate the effectiveness of the proposed method, which validates the second-order accuracy in solution approximations and the optimal linear complexity of computing time.

Mean-field sparse optimal control.

We introduce the rigorous limit process connecting finite dimensional sparse optimal control problems with ODE constraints, modelling parsimonious interventions on the dynamics of a moving population divided into leaders and followers, to an infinite dimensional optimal control problem with a constraint given by a system of ODE for the leaders coupled with a PDE of Vlasov-type, governing the dynamics of the probability distribution of the followers. In the classical mean-field theory, one studies the behaviour of a large number of small individuals freely interacting with each other, by simplifying the effect of all the other individuals on any given individual by a single averaged effect. In this paper, we address instead the situation where the leaders are actually influenced also by an external policy maker, and we propagate its effect for the number N of followers going to infinity. The technical derivation of the sparse mean-field optimal control is realized by the simultaneous development of the mean-field limit of the equations governing the followers dynamics together with the Γ-limit of the finite dimensional sparse optimal control problems.

Mean-Field Optimal Control

We introduce the concept of {\it mean-field optimal control} which is the rigorous limit process connecting finite dimensional optimal control problems with ODE constraints modeling multi-agent interactions to an infinite dimensional optimal control problem with a constraint given by a PDE of Vlasov-type, governing the dynamics of the probability distribution of interacting agents. While in the classical mean-field theory one studies the behavior of a large number of small individuals {\it freely interacting} with each other, by simplifying the effect of all the other individuals on any given individual by a single averaged effect, we address the situation where the individuals are actually influenced also by an external {\it policy maker}, and we propagate its effect for the number $N$ of individuals going to infinity. On the one hand, from a modeling point of view, we take into account also that the policy maker is constrained to act according to optimal strategies promoting its most parsimonious interaction with the group of individuals. This will be realized by considering cost functionals including $L^1$-norm terms penalizing a broadly distributed control of the group, while promoting its sparsity. On the other hand, from the analysis point of view, and for the sake of generality, we consider broader classes of convex control penalizations. In order to develop this new concept of limit rigorously, we need to carefully combine the classical concept of mean-field limit, connecting the finite dimensional system of ODE describing the dynamics of each individual of the group to the PDE describing the dynamics of the respective probability distribution, with the well-known concept of $\Gamma$-convergence to show that optimal strategies for the finite dimensional problems converge to optimal strategies of the infinite dimensional problem.

Optimal Path Following for Differentially Flat Robotic Systems Through a Geometric Problem Formulation

Path following deals with the problem of following a geometric path with no predefined timing information and constitutes an important step in solving the motion-planning problem. For differentially flat systems, it has been shown that the projection of the dynamics along the geometric path onto a linear single-input system leads to a small dimensional optimal control problem. Although the projection simplifies the problem to great extent, the resulting problem remains difficult to solve, in particular in the case of nonlinear system dynamics and time-optimal problems. This paper proposes a nonlinear change of variables, using a time transformation, to arrive at a fixed end-time optimal control problem. Numerical simulations on a robotic manipulator and a quadrotor reveal that the proposed problem formulation is solved efficiently without requiring an accurate initial guess.

Augmented Lagrangian Method for One Dimensional Optimal Control Problems Governed by Delay Differential Equation

In this research, numerical solutions of continuous optimal control problems governed by linear damping evolution with delay and real coefficients are presented. The necessary conditions obtained from the knowledge of calculus of variation for optimal control problem constrained by delay differential equation is a linear two-point boundary value problem involving both delay and advance terms. Clearly, this coupling that exists between the state variable and the control variable is not amenable to analytical solution hence a direct numerical approach is adopted. We propose an augmented discretized continuous algorithm via quadratic programming, which is capable of handling optimal control problems constrained by delay differential equations. The discretization of the problem using trapezoidal rule (a one step second order numerical scheme) and Crank-Nicholson with quadratic formulation amenable to quadratic programming technique for solution of the optimal control problems are considered. A control operator (penalized matrix), through the augmented Lagrangian method, is constructed. Important properties of the operator as regards sequential quadratic programming techniques for determining the optimal point are shown.

Implementation of Dynamic Programming for $n$-Dimensional Optimal Control Problems With Final State Constraints

Many optimal control problems include a continuous nonlinear dynamic system, state, and control constraints, and final state constraints. When using dynamic programming to solve such a problem, the solution space typically needs to be discretized and interpolation is used to evaluate the cost-to-go function between the grid points. When implementing such an algorithm, it is important to treat numerical issues appropriately. Otherwise, the accuracy of the found solution will deteriorate and global optimality can be restored only by increasing the level of discretization. Unfortunately, this will also increase the computational effort needed to calculate the solution. A known problem is the treatment of states in the time-state space from which the final state constraint cannot be met within the given final time. In this brief, a novel method to handle this problem is presented. The new method guarantees global optimality of the found solution, while it is not restricted to a specific class of problems. Opposed to that, previously proposed methods either sacrifice global optimality or are applicable to a specific class of problems only. Compared to the basic implementation, the proposed method allows the use of a substantially lower level of discretization while achieving the same accuracy. As an example, an academic optimal control problem is analyzed. With the new method, the evaluation time was reduced by a factor of about 300, while the accuracy of the solution was maintained.

Nonholonomic motion planning for a free-falling cat using spline approximation

An optimal motion planning of a free-falling cat based on the spline approximation is investigated. Nonholonomicity arises in a free-falling cat subjected to nonintegrable velocity constraints or nonintegrable conservation laws. The equation of dynamics of a free-falling cat is obtained by using the model of two symmetric rigid bodies. The control of the system can be converted to the motion planning problem for a driftless system. A cost function is used to incorporate the final errors and control energy. The motion planning is to determine control inputs to minimize the cost function and is formulated as an infinite dimensional optimal control problem. By using the control parameterization, the infinite dimensional optimal control problem can be transformed to a finite dimensional one. The particle swarm optimization (PSO) algorithm with the cubic spline approximation is proposed to solve the finite dimension optimal control problem. The cubic spline approximation is introduced to realize the control parameterization. The resulting controls are smooth and the initial and terminal values of the control inputs are zeros, so they can be easily generated by experiment. Simulations are also performed for the nonholonomic motion planning of a free-falling cat. Simulated experimental results show that the proposed algorithm is more effective than the Newtoian algorithm.

Reentry Trajectory Optimization Based on Differential Evolution

Reentry trajectory optimization is a multi-constraints optimal control problem which is hard to solve. To tackle it, we proposed a new algorithm named CDEN(Constrained Differential Evolution Newton-Raphson Algorithm) based on Differential Evolution(DE) and Newton-Raphson. We transform the infinite dimensional optimal control problem to parameter optimization which is finite dimensional by discretize control parameter. In order to simplify the problem, we figure out the control parameter’s scope by process constraints. To handle constraints, we proposed a parameterless constraints handle process. Through comprehensive analyze the problem, we use a new algorithm integrated by DE and Newton-Raphson to solve it. It is validated by a reentry vehicle X-33, simulation results indicated that the algorithm is effective and robust. Keywords—reentry vehicle, trajectory optimization, constraint optimal, differential evolution.

A Gait Generation Method for the Compass-type Biped Robot based on Discrete Mechanics

AbstractIn this paper, we consider a new approach based on discrete mechanics to a gait generation problem for the compass-type biped robot. First, both continuous-time and discrete-time models of the compass-type biped robot are derived. We next formulate a discrete gait generation problem for the discrete compass-type biped robot as a finite dimensional nonlinear optimal control problem, and solve it by using the sequential quadratic programming. Then, we propose a transformation method of a discrete-time input into continuous one and apply it to gait generation for the continuous compass-type biped robot. Some simulations are also shown in order to verify the effectiveness of our approach.

Optimal Exact Path-Following for Constrained Differentially Flat Systems

AbstractWe propose a dynamic optimization approach to calculate optimal feedforward controls for exact path-following problems of differentially flat systems. In addition to the derivation of a small dimensional optimal control problem, we provide easily checkable conditions on the existence of inputs guaranteeing that a given path is exactly followable in the presence of constraints on states and inputs. Our approach is based on the projection of the feedforward controlled nonlinear MIMO dynamics along a geometric path onto a linear single-input system in Brunovský normal form. The presented results indicate how the computation of admissible trajectories for set-point changes can be simplified by relying on so called steady state consistent paths. The set-point change of a Van der Vusse reactor is considered as an example.

Corrigendum to “A numerical scheme for two dimensional optimal control problems with memory effect” [Comput. Math. Appl. 59 (2010) 1630–1636

The author presents in below the necessary corrections on page 1631 of the original article ‘‘A numerical scheme for two-dimensional optimal control problems with memory effect’’ [Comput. Math. Appl. 59 (2010) 1630-1636]: – The text immediately after Equation 4 should read: ‘‘and satisfying the terminal condition x(a) = c. Here t denotes the time, x(t) and u(t) are a nx × 1 state and nu × 1 control vectors (not necessarily in same dimension), f and g are a scalar and a nx× 1 vector functions, aDαt x is the left RLFD of order α of xwith respect to t , and c is a given vector.’’ instead of ‘‘and satisfying the terminal conditions x(a) = c and x(b) = d. Here t denotes the time, x(t) and u(t) are a nx × 1 state and nu× 1 control vectors (not necessarily in same dimension), f and g are a scalar and a nx× 1 vector functions, aDαt x is the left RLFD of order α of xwith respect to t , and c , d are given vectors.’’ – The text of the first sentence of the paragraph after Equation 10 should read: ‘‘The equations in (7)–(9) describe the necessary conditions in terms of a Hamiltonian for the FOCP defined above.’’ instead of ‘‘The equations in (7)–(10) describe the necessary conditions in terms of a Hamiltonian for the FOCP defined above.’’

Pattern formation, spatial externalities and regulation in coupled economic–ecological systems

We study the management of ecosystems in which interacting state variables that are affected by management decisions diffuse in space. We study pattern formation first in a setting where economic agents maximize myopic profits and ignore spillovers onto agents at other sites, and second in the context of a social planner who internalizes these spillovers. We compare patterns under these two institutions and study how the differences in the patterns generated by these two institutions impact the design of regulatory schemes to internalize the externalities. At the same time we introduce new methods based on Fourier series type decompositions that enable us to turn an intractable infinite dimensional optimal control problem into a countable number of finite dimensional optimal control problems. As far as we know these methods are new to the economics literature. Our theory is applied to the management and regulation of a semi-arid system.

Optimal motion planning for a rigid spacecraft with two momentum wheels using Quasi-Newton method

An optimal motion planning scheme based on the quasi-Newton method is proposed for a rigid spacecraft with two momentum wheels. A cost functional is introduced to incorporate the control energy, the final state errors and the constraints on states. The motion planning for determining control inputs to minimize the cost functional is formulated as a nonlinear optimal control problem. Using the control parametrization, one can transform the infinite dimensional optimal control problem to a finite dimensional one that is solved via the quasi-Newton methods for a feasible trajectory which satisfies the nonholonomic constraint. The optimal motion planning scheme was applied to a rigid spacecraft with two momentum wheels. The simulation results show the effectiveness of the proposed optimal motion planning scheme.

Reduced Basis Method for Optimal Control of Unsteady Viscous Flows

In this article we discuss the reduced basis method (RBM) for optimal control of unsteady viscous flows. RBM is a reduction method in which one can achieve the versatility of the finite element method or another for that matter and gain significant reduction in the number of degrees of freedom. The essential idea in this method is to define a reduced order subspace spanned by few basis elements and then obtain the solution via a Galerkin projection. We present several ways to define this subspace. Feasibility of the approach is demonstrated on two boundary control problems in cavity and wall bounded channel flows. Control action is effected through boundary surface movement on part of the solid wall. Application of RBM to the control problems leads to finite dimensional optimal control problems which are solved using Newton's method. Through computational experiments we demonstrate the feasibility and applicability of the reduced basis method for control of unsteady viscous flows.

Infinite dimensional optimal control theory

One of the milestones in modern optimal control theory is the, Pontryagin Maximum Principle, which was firstly established by Pontryagin and his colleagues in later 50s for finite dimensional optimal control problems. In this survey paper, we revisit some extensions of the maximum principle in infinite dimensional spaces, developed via the infinite dimensional spike variation technique

An Evolutionary Algorithm Approach For Solving Optimal Control Problems

An approach to the approximate solution of general control problems, exploiting evolutionary global search and optimization techniques, is proposed. According to the methodology developed in this paper, controls are approximated with the help of appropriately chosen parameter vectors and substituted in system equations. Alternatevly, the phase coordinates may be parameterised, especially, when it is possible to represent controls as single valued functions of the states. In both cases the components of the parameter vectors are regarded as variables of a performance index based goal function that is to be minimized with respect to the system constraints. Such an approach enables modeling and solution of a wide class of optimal control problems, arising in engineering practice, within a unified framework of constrained optimization techniques, including evolutionary algorithms for global optimization and multiobjective control, and utilization of parallel processing to alleviate the computational burden in high dimensional optimal control problems.

Should controls be eliminated while solving optimal control problems via direct methods?

Direct methods of solving optimal control problems include techniques based on control discretization, where the control function of time is parameterized, and collocation, where both the control and state functions of time are parameterized. A recently introduced direct approach of solving optimal control problems via differential inclusions parameterizes only the state, and constrains the state rates to lie in a feasible hodograph space. In this method, the controls, which are just artifacts used to parameterize the feasible hodograph space, are completely eliminated from the optimization process. Explicit and implicit schemes of control elimination are discussed. Comparison of the differential inclusions method is made to collocation in terms of number of parameters, number of constraints, CPU time required for solution, and ease of calculation of analytical gradients. A minimum time-to-climb problem for an F-15 aircraft is used as an example for comparison. For a special class of optimal control problems with linearly appearing bounded controls, it is observed that the differential inclusion scheme is better in terms of number of parameters and constraints. Increased robustness of the differential inclusion methodology over collocation for the Goddard problem with singular control as part of the optimal solutions is also observed. Background T HE most precise approach to solve optimal control problems is the variational1 approach based on Pontryagin's minimum principle.2 This is an indirect approach as it involves solving the necessary conditions of optimality associated with the infinite dimensional optimal control problem rather than optimizing the cost of a finite dimensional discretization of the original problem directly. This method requires advanced analytical skills and to generate numerical solutions of the resulting two-point boundary-value problem is highly nontrivial. The controls are eliminated in the indirect method using the minimum principle. Thus, the optimal control is, in general, a nonlinear function of the state and costate variables. The most important application of the indirect method is the generation of benchmark solutions. Usually, good convergence is achieved only with excellent initial guesses for the nonintuitive costates. Additionally, the switching structure has to be guessed correctly in advance. For rapid trajectory prototyping, the safest approaches are the direct methods.3 These methods rely on a finite dimensional discretization of the optimal control problem to a nonlinear programming problem. Even though these methods do not enjoy the high precision and resolution of indirect methods, their convergence robustness makes them the method of choice of most practical applications. Moreover, these methods do not require the advanced mathematical skills necessary to pose and solve the variational problem.

Dynamic Optimization in the Design and Scheduling of Multiproduct Batch Plants

Dynamic batch processing provides additional transient operating freedom, that can stretch the limits of profitability under strict market, facility, and time constraints. This paper incorporates dynamic processing conditions for products in a multiproduct batch plant, as opposed to fixing the process by recipes, in the broader context of equipment design, production planning, scheduling, and inventory considerations. The objective is a general function of fixed design costs, operating costs, production revenues, etc. Decisions include stage processing times for products, transient stage operating policies, continuous design parameters, production capacity, and production schedules. The infinite dimensional optimal control problem for each operation is solved using collocation over finite time elements (Cuthrell, J. E.; Biegler, L. T. Comp. Chem. Eng. 1989, 13, 49. Logsdon, J. S.; et al. Trans. Inst. Chem. Eng. 1990, 68A, 434). Scheduling, with its combinatorial complexity, is addressed in the scope of flowshop plants for specific transfer policies using the aggregated scheduling model in Birewar, D. B.; Grossmann, I. E. Ind. Eng. Chem. Res. 1989a, 28, 1333; Comp. Chem. Eng. 1989b, 13, 141.